Cylindrical Micelles in Rigid-Flexible Diblock Copolymers
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CYLINDRICAL MICELLES IN RIGID-FLEXIBLE DIBLOCK COPOLYMERS D.R.M Williams and G.H. Fredrickson Departments of Materials and Chemical & Nuclear Engineering University of California, Santa Barbara, CA 93106, U.S.A.
ABSTRACT We present a theoretical study of a melt of diblock copolymers consisting of a rigid rod and a flexible tail. It is shown that in addition to the lamellax phases previously discussed, there also exist phases of "hockey puck" micelles, where the rods are packed axially into cylinders. These phases occupy most of the phase diagram previously thought to consist of monolayer lamellae. It is argued that spherical micelles probably do not exist.
Introduction In a recent paper [1], Semenov and Vasilenko initiated a study of melts of diblock copolymers consisting of a flexible tail connected to a rigid rod. They assumed that the tail and rod segments were incompatible with the degree of incompatibility described by via a Flory parameter XS. In general one expects xS to be large at room temperature, because of the large chemical differences between stiff rods and flexible chains. At low enough XS values the the system exhibits a nematic phase (provided the rod volume fraction is sufficiently large) which, as the interaction parameter is increased, changes to a smectic phase. As XS is further increased the smectic phase was predicted to undergo various other changes - first to a complete monolayer regime (figure (1)) where the chains are expelled completely from the rigid rod monolayers, and then to a bilayer lamellar phase (figure (1)). If the volume occupied by the flexible tails is small, then the phases studied in [1] are probably the most stable. In this paper we study the possibility of forming discrete micellar phases. One expects these to be stable in the limit where the flexible tail volume fraction is large. Here we propose the existence of cylindrical disk-like micelles (see figure 2). These are micelles consisting of pieces of lamellae cut into cylinders. Their main advantage relative to lamellae consists of enabling the chains grafted onto their top and bottom surfaces to fan out into a larger region of space (ultimately increasing as r 3 ) and hence stretching less. Their main disadvantage is the creation of an extra "curved" surface, for which they pay a surface energy penalty. It is the balance between these two terms that decides whether they are more favoured than lamellae. In this paper we shall use the terms puck and cylinder interchangeably, although "puck" reflects the rather flat character of the micelles more accurately. The nematic and smectic phases We begin with some definitions. We set Boltzmann's constant equal to unity throughout, and express all free energies in units of kBT. We consider rods of length L and cross sectional area d2 where d < L. The flexible part of the blocks is assumed to consist of N segments of volume v with mean square separation between adjacent segments of 6a 2 in the unperturbed melt state. We use the same dimensionless parameters as [1], namely A = Nv
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